If unmarried parents never establish custody within 5 yrs can the mother relocate with the child?

yes as long as no case is pending in court.

An unmarried mother has sole legal custody of her child until the father takes the necessary steps to establish his paternity and obtain a custody order. He can request a visitation schedule through the court and the court will enter a child support order. If he does not take those steps and the mother relocates it will be more difficult for him to establish his rights.